I do not disagree with kipling's post. However, I would like to add a couple of points.
First, I think that most people who watched would simply have their own preexisting biases confirmed. They would watch the execution through their own biased lenses and would therefore tend to see what they wanted. A proponent of capital punishment would probably watch a lethal injection and think “that’s a really easy death for someone who committed a heinous crime.” An opponent would be moved by the fact that a person was being killed by the government and would have their inclination to reject that act strengthened.
Second, I would argue that the way in which the execution was presented would have a great deal to do with how people perceived it. For example, imagine if the details of the crime for which the person was being executed were emphasized before the execution. Even more, imagine if gory pictures from the crime scene were presented. Alternatively, imagine if the broadcaster chose to do a retrospective of the life of the condemned, focusing on problems with their upbringing and then, perhaps, looking at how they had apparently reformed while in prison. These would be very different presentations and would affect people in very different ways.
Whether live broadcasts of prisoner executions would fundamentally change American attitudes towards capital punishment is essentially unknowable. Assuming viewership was not somehow mandatory and enforceable, people could simply choose not to view the execution. Those who did choose to view the execution would likely be divided into the same categories that already divide many Americans. Some opponents of capital punishment would watch as a way of validating their opposition to state-sponsored killing. Some supporters of capital punishment would make a point of watching for the pure experience of watching a condemned human being executed. There is an audience for such broadcasts. Public executions in Saudi Arabia, which involve beheadings rather than the more humane and philosophically-tortured process of killing via lethal injection, are well-attended events, with thousands of people cheering on the execution of a sentence against a man or woman accused of a crime against Islam. The less said about the underground audience for snuff films the better, but the point remains that there is among any population those with a morbid fascination in viewing deaths. As with capital punishment opponents who feel morally compelled to view such a broadcast, there would certainly be supporters of capital punishment who would feel morally obligated to view the broadcast.
It is highly likely that viewing executions would influence some supporters of capital punishment to change their views, but they would probably not be representative of the majority. Supporters of capital punishment are under no illusions regarding the morbidity of the enterprise and the ugliness of the act; they remain committed to the notion that execution of a convicted prisoner is the appropriate penalty for certain crimes, especially murder. Viewing the execution can provide closure for some and reinforce a sense of justice for others. In a way, the introduction of lethal injection probably made capital punishment even more palatable an option for supporters of capital punishment. The concept of lethal injection in itself is a reflection of a people viscerally torn with the notion of state-sponsored executions while remaining convinced of the appropriateness of capital punishment for those who have committed particularly heinous crimes. Whether the broadcast of executions by firing squad, hanging, electrocution, or asphyxiation by gas would have a greater impact than execution by lethal injection is a fair question, but, again, both sides in the debate recognize that a human being is being put to death.